Grover - 1990 - Nidt ( K R )( i = 1 ..., n , 2a R !

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Vol.136,No.6TheAmericanNaturalistDecember1990RESOURCECOMPETITIONINAVARIABLEENVIRONMENT:PHYTOPLANKTONGROWINGACCORDINGTOMONOD'SMODELJAMESP.GROVER*MaxPlanckInstituteofLimnology,DepartmentofPhysiologicalEcology,P.O.Box165,D-2320Plon,WestGermanySubmittedNovember4,1988;AcceptedJuly26,1989Ecologicalcommunitiesareextraordinarilycomplex.Theyoftencontainhun-dredsofspecies,whichmayinteractbydiversemeans.Oneimportantinteractionisresourcecompetition,theconsumptionofsharedresourcesbydifferentspe-cies.Ifspeciesdifferintheirabilitytoexploitsharedresources,thenthosemostabletoexploitresourceswillceterisparibus,competitivelyexcludeotherspecies.Aparadoxisthusposed.Inmanycommunities,especiallythoseofprimarypro-ducers,thenumberoflimitingresourcesseemssmall.Ifeachspeciesisasuperiorcompetitorforasingleresource,thenthenumberofspecieswilleventuallyequalthenumberofresources;ifaspeciescanbeasuperiorcompetitorformorethanoneresource,thenthenumberofspecieswillbeevenfewer.Why,then,doecologicalcommunitiescontainsomanyspecies?Thisstudyfocusesononecommunity,phytoplankton,andonepossiblean-swer.Hutchinson(1961)notedthatforsuperiorcompetitorstoexcludeinferiorones,aconstantenvironmentthatallowsequilibriumtobereachedmustbeassumed.Iftheenvironmentvariesintime,thencompetitiveexclusionisnotcertain(Hutchinson1961;ArmstrongandMcGehee1980).Aspectrumofvariabil-ityinphysical,chemical,andbiologicalfactorsisfoundinopen-waterhabitats,forcingustoquestionwhetherphytoplanktoneverapproachequilibrium.Some(e.g.,Harris1986)suggestnotonlythatvariabilityallowscoexistenceofmanyspeciesbutthatanypatternsinphytoplanktonthatmightresultfromresourcecompetitionaredestroyedbyvariability.Muchisknownaboutresourcecompetitionamongphytoplanktonfordissolvednutrientsasresourcesinequilibriumsystems.TheMonod(1950)modelisarea-sonabledescriptionofalgalgrowthasafunctionofdissolved-nutrientconcentra-tion(Tilmanetal.1982;P.KilhamandHecky1988).Inthismodel,thepercapitagrowthrateofanalgalpopulation(ofcelldensityN)isdNINdt=rRI(K+R),(1)whereRisextracellularresourceconcentration,risthemaximalgrowthrate,*Presentaddress:CentreforPopulationBiology,ImperialCollegeatSilwoodPark,Ascot,Berk-shireSL57PY,England.Am.Nat.1990.Vol.136,pp.771-789.?1990byTheUniversityofChicago.0003-0147/90/3606-0001$02.00.Allrightsreserved.Thiscontentdownloadedfrom128.135.012.127onNovember27,201613:05:11PMAllusesubjecttoUniversityofChicagoPressTermsandConditions(http://www.journals.uchicago.edu/t-and-c). 772THEAMERICANNATURALISTandKistheresourceconcentrationatwhichgrowthishalfthemaximum.Inthismodel,algalgrowthrespondsinstantaneouslytoextracellularresourceconcentra-tion;intracellularstorageoftheresourcedoesnotoccur.Whennspeciescompeteforthesameresourceinacontinuousculture(Monod1950)withadilutionrateofDandaconstantsupplyofresourceataconcentrationofR?,dN.r.R=__DY=1.I(aNidt(K;+R)(i=1...,n,2adR=D(R?-R)-EN1r+R(2b)dtZ.Y1(Kj+R)'(bwhereYiareyieldcoefficientsdescribingtheconversionofresourceintoalgalcells.Insuchaculture,thecontinuousflowofmediumimposesconstant,density-independent,andnonselectivepercapitaloss(onresidentpopulations)equaltoD.Attheequilibriumbetweenlossandgrowthrates,onlyonespeciespersists,thesuperiorcompetitor.Ithasthelowerresourcerequirementatequilibrium(R1j,whengrownintheabsenceofcompetitors(Hsuetal.1977;Tilman1977):R!*=(rDKD)-D)'(3)3LaboratorystudieshaveverifiedtheutilityofR*asapredictorofthecompetitiveoutcomeatequilibrium(Tilman1977;HolmandArmstrong1981;TilmanandSterner1984;S.Kilham1986).Howdothepredictionsofthistheorychangewhenequilibriumispreventedbyvariabilityinresourcesupply?Inthiscase,equation(2)becomesanonautono-moussystemwithR?=R0(t).Fortwocompetitorsandaresourceinputthatvariessinusoidally(RO=R+bsinwt,whereRistheaverageinputresourceconcentration;b,amplitude;w,angularfrequency,assumingthatbrl.DashedlineD,thedilutionrateusedinsimulations.WherelineDintersectsthegrowthfunctionforaspecies,thevalueoftheabscissaisthatspecies'equilibriumrequire-mentforphosphorus(R*).NotethatR*Vo)Wc,C,KEl000"c00'ftr1r41HL4,Hz>>azo~~~~~~~~~0E30:R>:.?:.<:?OOOozo.oooo